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Quadruple fixed point theorems for nonlinear contractions in partially ordered metric spaces. (English) Zbl 1264.47057
Let \((X,d,\leq)\) be a complete ordered metric space, \(g:X\to X\) and \(F:X^4\to X\) have the mixed \(g\)-monotone property (i.e., \(F\) is \(g\)-nondecreasing in the first and the third and \(g\)-nonincreasing in the second and the fourth variables) and satisfy the condition \[ d(F(x,y,z,w),F(u,v,r,t))\leq\phi(\frac14(d(gx,gu)+d(gy,gv)+d(gz,gr)+d(gw,gt))) \] whenever \(gx\leq gu\), \(gy\geq gv\), \(gz\leq gr\) and \(gw\geq gt\), where \(\phi:[0,\infty)\to[0,\infty)\) satisfies \(\phi(t)<t\) for \(t>0\) and \(\lim_{r\to t+}\phi(r)<t\) for each \(t>0\). Under some additional conditions, the authors prove that there exist \(x,y,z,w\in X\) such that \[ F(x,y,z,w)=gx,\quad F(x,w,z,y)=gy, \quad F(z,y,x,w)=gz, \quad F(z,w,x,y)=gw, \] i.e., \(F\) and \(g\) have a so-called quadruple coincidence point. The uniqueness of such a point is also obtained under certain conditions. Some very easy examples are presented.

MSC:
47H10 Fixed-point theorems
46J10 Banach algebras of continuous functions, function algebras
54H25 Fixed-point and coincidence theorems (topological aspects)
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